Finite Group Databasebeta

← #10692521

Subgroups — #10692521

Order 768.

582 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78514C211yesmaximal, characteristic, complemented
2#73426C211yesmaximal, characteristic
3#70826C211yesmaximal, characteristic, complemented
4#70804C211yesmaximal, characteristic, complemented
5#69860C211yesmaximal, characteristic, complemented
6C3×#1300C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#9387—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×#875C2211yescharacteristic
11C2×#872C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C3×#376C2211yescharacteristic
15Dic3×M16—21no—
16#4940—21no—
17#4939—21no—
18#4926C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4708—22no—
21#4689—22no—
22#4673C411yescharacteristic
23#4672C411yescharacteristic
24#4671C411yescharacteristic
25#4670—21no—
26#4665—21no—
27#4656C411yescharacteristic
28#4469—21no—
29#4468—21no—
30#4464—22no—
31#4464—22no—
32#2873—31no—
33#2610—31no—
34#1527—31no—
35#1527—31no—
36#1464—31no—
37#1300—31no—
38#1300D311yescharacteristic, complemented
39C22×D12—21no—
40C22×D12C2×C411yescharacteristic
41C22×C4×D3D411yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4C2×C411yescharacteristic
53C2×(C2×D3)⋊C4D411yescharacteristic
54C2×C4⋊Dic3—21no—
55C2×Dic3⋊C4C2×C411yescharacteristic
56C2×Dic3⋊C4D411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C2×#208—21no—
59D3×M16—24no—
60D3×M16—24no—
61C2×ΓL(1,𝔽25)—22no—
62C2×D3×C8—22no—
63#875—21no—
64#875—21no—
65#875—21no—
66#875—21no—
67#872—21no—
68#872—21no—
69#872—21no—
70#872—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97D412yescomplemented
75C3×C4×C8—21no—
76C3×#95—21no—
77C3×#95C2×C411yescharacteristic
78C3×#95C2×C411yescharacteristic
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#815—22no—
83C8×Dic3—22no—
84C4⋊(C3⋊C8)—21no—
85#803—21no—
86#803D412yescomplemented
87C4×C3⋊C8—21no—
88C22×M16—61no—
89C2×#124—31no—
90C2×#117—31no—
91#432—31no—
92#432C2×D311yescharacteristic
93#405—62no—
94#405—62no—
95C4×M16—31no—
96C4×M16—61no—
97C4×M16C2×D311yescharacteristic
98#395—31no—
99#394—31no—
100#393—31no—
101#392—31no—
102#390—61no—
103#388—61no—
104#388—61no—
105#386—61no—
106#376C2×D311yescharacteristic
107#329—62no—
108#329—62no—
109#324—61no—
110#323—61no—
111C23×D3—21no—
112C23×D3C22⋊C411yescharacteristic
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—21no—
119C22×Dic3C22⋊C411yescharacteristic
120C22×Dic3C22⋊C411yescharacteristic
121C22×Dic3C22⋊C411yescharacteristic
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—42no—
125C2×D12—42no—
126C2×D12—81no—
127C2×D12—81no—
128C2×C4×D3—22no—
129C2×C4×D3—21no—
130C2×C4×D3—21no—
131C2×C4×D3—22no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—81no—
135C3×M16—22no—
136C3×M16—22no—
137C2×C3×C8—22no—
138C2×C3×C8—41no—
139C3×C42—42no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144(C2×D3)⋊C4—21no—
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—21no—
147(C2×D3)⋊C4—21no—
148(C2×D3)⋊C4—42no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—42no—
154(C2×D3)⋊C4—81no—
155(C2×D3)⋊C4—81no—
156C4⋊Dic3—81no—
157Dic3⋊C4—21no—
158Dic3⋊C4—21no—
159Dic3⋊C4—21no—
160Dic3⋊C4—21no—
161C4×Dic3—21no—
162C4×Dic3—21no—
163#208—22no—
164#208—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167ΓL(1,𝔽25)—28no—
168D3×C8—28no—
169C22×Q8—31no—
170C22×D4—31no—
171C22×D4—61no—
172C23×C4—31no—
173C2×M16—21no—
174C2×M16—62no—
175C2×M16—64no—
176C2×M16—64no—
177C2×M16—61no—
178C22×C8—62no—
179#124—61no—
180#124—61no—
181#124—61no—
182#124—61no—
183#117—61no—
184#117—61no—
185#117—61no—
186#117—61no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—31no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C42—21no—
199C2×C42—31no—
200C2×C42C22×D311yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—61no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205#97—21no—
206#97—32no—
207#97—62no—
208#97—61no—
209#97D1212yescomplemented
210C4×C8—21no—
211C4×C8—62no—
212C4×C8—61no—
213#95—21no—
214#95—61no—
215#95—61no—
216#95C4×D311yescharacteristic
217#95C4×D311yescharacteristic
218C23×C3C2×C22⋊C411yescharacteristic
219C22×D3—21no—
220C22×D3—21no—
221C22×D3—41no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3#10412yes—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—22no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12311yescharacteristic
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3#10412yes—
257D12—81no—
258D12—81no—
259D12—84no—
260C4×D3—22no—
261C4×D3—22no—
262C4×D3—24no—
263Dic6—81no—
264Dic6—81no—
265C3×C8—24no—
266C3×C8—81no—
267C3⋊C8—24no—
268C3⋊C8—81no—
269C24—31no—
270C24—61no—
271C2×Q8—61no—
272C2×Q8—61no—
273C2×Q8—241no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—122no—
277C2×D4—122no—
278C2×D4—241no—
279C2×D4—241no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4C2×C4×D311yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293M16—22no—
294M16—22no—
295M16—68no—
296M16—62no—
297M16—62no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—68no—
301C2×C8—62no—
302C2×C8—121no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—241no—
308C22⋊C4—61no—
309C22⋊C4—61no—
310C22⋊C4—61no—
311C22⋊C4—61no—
312C22⋊C4—122no—
313C22⋊C4—122no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—241no—
319C22⋊C4—241no—
320C42—42no—
321C42—61no—
322C42—61no—
323C42C2×D1211yescharacteristic
324C42#23511yescharacteristic
325C42D3×D411yescharacteristic
326C42#23911yescharacteristic
327C22×C3—21no—
328C22×C3—21no—
329C22×C3#38611yescharacteristic
330C22×C3C2×#10411yescharacteristic
331C22×C3C2×#10411yescharacteristic
332C2×D3—22no—
333C2×D3—41no—
334C2×D3—41no—
335C2×D3—42no—
336C2×D3—81no—
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—81no—
342C2×D3—81no—
343C2×D3—81no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—42no—
351C3×C4—81no—
352Dic3—22no—
353Dic3—41no—
354Dic3—41no—
355Dic3—81no—
356C23—32no—
357C23—61no—
358C23—61no—
359C23—121no—
360C23—121no—
361C23—121no—
362C23—121no—
363C23—121no—
364C23—121no—
365C23—121no—
366C23—121no—
367C23—121no—
368C23D3×C22⋊C411yescharacteristic
369Q8—241no—
370Q8—241no—
371D4—241no—
372D4—241no—
373D4—244no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—32no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—64no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4C4×D1211yescharacteristic
398C2×C4#88211yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4#89311yescharacteristic
401C2×C4#89411yescharacteristic
402C2×C4#89511yescharacteristic
403C8—24no—
404C8—64no—
405C8—81no—
406C8—241no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3#188411yescharacteristic
410C2×C3#189611yescharacteristic
411C2×C3#191411yescharacteristic
412D3—42no—
413D3—161no—
414C22—21no—
415C22—21no—
416C22—62no—
417C22—122no—
418C22—121no—
419C22—121no—
420C22—241no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22—241no—
425C22—241no—
426C22—241no—
427C22—241no—
428C22#466511yescharacteristic
429C22D3×#10411yescharacteristic
430C22D3×#10411yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—62no—
439C4—81no—
440C4—121no—
441C4—121no—
442C4—241no—
443C3#938711yescharacteristic, complemented, Sylow
444C2—21no—
445C2—21no—
446C2—122no—
447C2—481no—
448C2#6875611yescharacteristic
449C2#6877711yescharacteristic
450C2#6904011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#386
prime radical#4671C4
cosocleC2×C3×C42C23
hypercenter#1300D3
hypocenterC3#9387
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #78514C2semidirect
3. #70826C2semidirect
4. #70804C2semidirect
5. #69860C2semidirect
6. C3×#1300C2semidirect
7. #67615C2semidirect
38. #1300D3semidirect
74. C3×#97D4semidirect
86. #803D4semidirect
209. #97D12semidirect
443. C3#9387semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#785142
#734262
#708262
#708042
#698602
C3×#13002
#676152
#93873